Algebraic Topology 17: Degree and Cellular Homology
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- čas přidán 7. 09. 2024
- Playlist: • Algebraic Topology
We introduce the notion of the degree of a map from S^n to S^n. As a nice application, we use degree to prove the Hairy Ball Theorem. Then we develop cellular homology, another homology theory equivalent to simplicial and singular homology. We show how to calculate the cellular homology of the torus and Klein bottle.
Presented by Anthony Bosman, PhD.
Learn more about math at Andrews University: www.andrews.ed...
In this course we are following Hatcher, Algebraic Topology: pi.math.cornel...
Amazing professor looking forward to next lecture also can we except a course on algebraic geometry??
AG is not on the schedule any time soon, but that would be awesome!
for the long exact sequence of relative homologies at 41:43, wouldn't the cellular homology be trivial if that sequence is exact?
came here to ask this also
very interesting lecture, Thank you Sir❤, I took this degree of map definition in differential topology for about 15 min with the examples provided. Let say if f: CP^2 to itself defined by the f([z_0:z_1:z_2])=[z_0z_1:z_1z_2:z_2z_1]...I think if it is of the form f([z_0:z_1:z_2])=[z^3_0:z^3_1:z^3_2] then its degree is deg f=3^2=9, So how do i find the degree of f([z_0:z_1:z_2])=[z_0z_1:z_1z_2:z_2z_1] and why?
Thanks!
Lectures are amazing. is it possible to upload it fast? like 2 or 3 lectures in a week? it will be very much helpful !
At 1:02:00, the klein bottle’s face e_2 is mapped to 2a + 0b and not to both simultaneously, because boundary(n cell) = sum of (degree of map times (n-1)-cell), right?
“So we’ve shown that all of these homology groups are really just direct sums of Z. But I haven’t told you what the maps d_n are yet”
“Ok, what are the d n?”
“deez nuts haha gottem”
Only a total n-cell would make a joke like that.
At 1:03:42 don’t we have to set the relation equal to 0??
Yes, since the group is abelian it is standard to use 0 for additive identity. I erred in writing 1, instead.
Thanks for clarifying professor I was confused for 10 minutes or so 😅😅
So if it were a multiplicative group we could have used 1??
@@ompatel9017 as long as you're writing a^2 instead of 2a. 😉